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DMAX – Matrix of Dominant Distances in a Graph ** Milan Randi

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DMAX – Matrix of Dominant Distances in a Graph ** Milan Randi MATCH MATCH Commun. Math. Comput. Chem. 70 (2013) 221-238 Communications in Mathematical and in Computer Chemistry ISSN 0340 - 6253 * DMAX – Matrix of Dominant Distances in a Graph ** Milan Randi National Institute of Chemistry, Ljubljana Hajdrihova 19, Slovenia [email protected] (Received May 23, 2012) Abstract We are introducing a novel distance-type matrix for graphs, referred to as DMAX, which is constructed from the distance matrix of a graph be selecting for each row and each column only the largest distances. This matrix can be viewed as opposite to the adjacency matrix, which can be constructed from the distance matrix be selecting for each row and each column only the smallest distances, which correspond to adjacent vertices. We have illustrated the novel matrix for linear n-alkanes and n-star graphs, as well as the nine isomers of heptane. In the case of isomers, the matrix appears very sensitive on the branching pattern of molecular graphs. Therefore it may be of interest for characterizing of molecular shapes. On the other hand, in view that selected invariants of DMAX for structurally related molecules could be significantly different, it appears that such invariants of DMAX can be of interest in testing graphs for isomorphism. 1. Introduction Graphs have been of interest in chemistry for quite a while, primarily because selected graph invariants can serve as molecular descriptors. Graph invariants are any of mathematical properties of graphs that are independent of the numbering of graph vertices or of the geometrical representation of a graph. One class of graph invariants follows as matrix invariants *This contribution is dedicated to Professor Ivan Gutman, on the occasion of his 65th anniversary, for his scientific accomplishments as outstanding pioneer of Mathematical Chemistry, for his contributions to Graph Theory and Chemical Graph Theory, and his leadership as the current Editor of MATCH Communications in Mathematical and Computer Chemistry. **Emeritus, Dept. of Mathematics & Computer Science, Drake University, Des Moines, IA, USA -222- from representations of graphs by matrices. The well-known and oldest graph matrices are: the adjacency matrix, introduced by Poincaré for characterizing of labyrinths [1]; and the distance matrix, introduced by Frank Harary [2]. In more recent time, with the growth of Chemical Graph Theory, additional graph matrices have been introduced, like: the Wiener matrix [3-7]; the path-Wiener matrix [8]; the Szeged matrix [9, 10] and the revised Szeged matrix [11]; the Distance/Distance matrix [12]; resistance-distance matrix [13, 14]; novel graph distance matrix based on viewing rows as points in n-dimensional space [15], as well as matrices the elements of which are not numbers, but subgraphs, such as the double invariants matrix [16] and the Ulam subgraph matrix [17]. For a more complete review on matrices in chemistry we direct readers to the book of Janeži et al. [18]. In this contribution we would like to introduce a novel distance-type graph matrix, which is characterized by its simple conceptual content as well as the simple constructional procedure. What makes this matrix in some respect different from most of the above mentioned matrices is that it is sensitive to the pattern of branching in a graph, at least for the tree graphs (acyclic graphs) that we consider in this contribution. So two apparently similar graphs can nevertheless have visibly different matrices, which will consequently have a number of fairly different graph invariants. While such matrices and their invariants are therefore to of limited and little interest in structure-property and structure-activity studies, because similar molecules will have dissimilar molecular descriptors, they may be of considerable interest for characterization of graph shapes and for investigating graph isomorphism. 2. Construction of the DMAX Matrix of a Graph We will illustrate the construction of the DMAX matrix on the molecular graph of 4-ethyl-2- methyl-heptane. We start by construction of the distance matrix for this graph, which is illustrated in Table 1. In Table 2 we show for each row of the distance matrix only its largest entries (distances). When this matrix is transposed one obtains a portion of the distance matrix showing the largest entries for each column. When one combines the matrix elements which are the largest in each row and in each column of the distance matrix and sets all missing matrix elements in the matrix to be zero, one obtains the DMAX matrix of 4-ethyl-2-methyl-heptane shown in Table 3. Formally the definition of the DMAX matrix is: -223- Let D be the distance matrix. Let Ri be the greatest element in the i-th row of D. Let Cj be the greatest element in the j-th column of D. Then I $ L()D ij if () Dij min,% Ri C j ()DMAX ij J L * % $ K0()min,if Dij Ri C j Table 1 Distance matrix of 4-ethyl-2-methyl-heptane 1 2 3 4 5 6 7 8 9 10 1 0 1 2 3 4 5 6 2 4 5 2 1 0 1 2 3 4 5 1 3 4 3 2 1 0 1 2 3 4 2 2 3 4 3 2 1 0 1 2 3 3 1 2 5 4 3 2 1 0 1 2 4 2 3 6 5 4 3 2 1 0 1 5 3 4 7 6 5 4 3 2 1 0 6 4 5 8 2 1 2 3 4 5 6 0 4 5 9 4 3 2 1 2 3 4 4 0 1 10 5 4 3 2 3 4 5 5 1 0 If in the above definition of DMAX one would replace the condition: greatest element by smallest non-zero element, then one would obtain the adjacency matrix. Thus, in a way, one can look at DMAX as the opposite of the adjacency matrix, the anti-adjacency matrix. From this point Table 2 The largest matrix elements in each row of the distance matrix of -4-ethyl-2- methyl-heptane (Table 1) 1 2 3 4 5 6 7 8 9 10 1 0 6 2 0 5 3 0 4 4 3 0 3 3 5 4 0 4 6 5 0 5 7 6 0 6 8 6 0 9 4 4 4 0 10 5 5 0 -224- Table 3 The DMAX for 2-methyl-4-ethyl-heptane 1 2 3 4 5 6 7 8 9 10 1 0 0 0 3 4 5 6 0 4 5 2 0 0 0 0 0 0 5 0 0 0 3 0 0 0 0 0 0 4 0 0 0 4 3 0 0 0 0 0 0 3 0 0 5 4 0 0 0 0 0 0 4 0 0 6 5 0 0 0 0 0 0 5 0 0 7 6 5 4 3 0 0 0 6 4 5 8 0 0 0 3 4 5 6 0 4 5 9 4 0 0 0 0 0 4 4 0 0 10 5 0 0 0 0 0 5 5 0 0 of view one can recognize the novel matrix as one of the more elementary graph matrices, ranking in importance probably next to the adjacency matrix and the distance matrix. It seems therefore of considerable interest to investigate the mathematical properties of this novel graph matrix. 3. Reconstruction of a Graph from DMAX The first and important question is whether the DMAX allows the reconstruction of the graph, because if it does, then this means that there is no loss of information on the graph and that no two graphs can have the same DMAX. Even though DMAX is defined for general graphs (acyclic or cyclic) we will confine our attention mostly to trees. It is known that in case of trees it is sufficient to know all the distances between its terminal vertices. Once the terminal vertices have been identified, one can find all the distances between terminal vertices. According to a theorem of Zaretskii [19] this information suffices for graph reconstruction. The question is whether from DMAX one can extract information on distances between all terminal vertices. Clearly the largest entries in DMAX will identify several terminal vertices, but one needs distances between all terminal vertices. In the case of 4-ethyl-2-methyl-heptane we can see from Table 3 that matrix elements (1,7) and (7,8) are the largest. This immediately suggests that the vertices 1, 7, 8 are terminal. Moreover, it is not difficult to deduce that graph of 2-methylheptane is a subgraph of the graph of 4-ethyl-2-methyl-heptane graph, which accounts for eight of ten carbon atoms of 2-methyl-4- ethylheptane. This information suffices to construct the part of the adjacency matrix belonging -225- to vertices 1-8. From this part of attempted reconstruction we found out that the distance between terminal vertices 1 and 2 is not included in DMAX, though it was possible to determine it. Hence, DMAX does not necessarily involve information on all distances between terminal vertices. From the remaining columns and rows of the DMAX matrix of 4-ethyl-2-methyl-heptane one finds that matrix elements (1,10), (7,10) and (8,10) are the largest entries. This points that vertex 10 is at the same distance vertices 1, 7, and 8. The only vertex at the same distance from these three vertices is vertex 4, which must be the site of ethyl group. This suffices to identify vertex 4 where the ethyl branch of 4-ethyl-2-methyl-heptane is attached to 1,2-dimethylheptane subgraph to make 4-ethyl-2-methyl-heptane – which completes the reconstruction of this particular graph.
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